Weyl-Schrödinger representations of Heisenberg groups in infinite dimensions

Abstract

A complexified Heisenberg matrix group ${\mathrm{H}}_{\mathbb{C}}$ with entries from an infinite-dimensional Hilbert space $H$ is investigated. The Weyl--Schrödinger type irreducible representations of ${\mathrm{H}}_{\mathbb{C}}$ on the space $L_{\chi }^2$ of square-integrable scalar functions is described. The integrability is understood under the invariant probability measure $\chi$ which satisfies an abstract Kolmogorov consistency conditions over the infinite-dimensional unitary group $U(\infty )$ irreducible acted on $H$. The space $L_{\chi }^2$ is generated by Schur polynomials in variables on Paley--Wiener maps over $U(\infty )$. Therewith, the Fourier-image of $L_{\chi }^2$ coincides with a space of Hilbert--Schmidt entire analytic functions on $H$ generated by suitable Fock space. Applications to linear and nonlinear heat equations over the group ${\mathrm{H}}_{\mathbb{C}}$ are considered.

29 pages